Chapter 1 Study Guide

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Table of Contents

Chapter 1 — Linear Equations in Linear Algebra

Covers §1.1–1.7 (§1.6 not assigned). Based on Lay, Linear Algebra and Its Applications, 6e. Every problem referenced here is transcribed in full with its recorded answer in 2160 Compendium.

Weighting

104 questions — the largest chapter by far, and 30 of them (28%) are true/false conceptual statements. Computation alone will not carry this chapter; the definitions have to be precise enough to judge a sentence.

§1.1 Systems of Linear Equations

Definition 1 (Core vocabulary).

A system is consistent if it has at least one solution, inconsistent if none. Two matrices are row equivalent if a sequence of elementary row operations turns one into the other. Row equivalent systems have the same solution set.

The three elementary row operations

Replacement — add a multiple of one row to another
Interchange — swap two rows
Scaling — multiply a row by a nonzero constant

All three are reversible. That reversibility is what makes row equivalence an equivalence relation, and it is the single most-tested idea in the §1.1 true/false problems.

Existence and uniqueness. A system is inconsistent exactly when an echelon form has a row $[\,0\ \cdots\ 0\ \mid b\,]$ with $b \neq 0$. If consistent, the solution is unique when there are no free variables, and infinite otherwise.

Tested as

Solve a $2\times2$ system by elimination — 1.1.1, 1.1.2, 1.1.13
Intersection of two lines / three planes — 1.1.3, 1.1.22
Read consistency off a reduced augmented matrix — 1.1.7, 1.1.10
Name the row operation that transforms one matrix into another — 1.1.39, 1.1.41
True/false on row operations and row equivalence — 1.1.27 – 1.1.31

Traps

“Row equivalent” is about the matrices, not the solution sets — two systems can share a solution set without being row equivalent. Elementary row operations are reversible, so no information is lost; a scaling by zero is not an elementary operation.

§1.2 Row Reduction and Echelon Forms

Definition 2 (Echelon vs. reduced echelon).

Echelon form: all nonzero rows above any zero rows; each leading entry strictly right of the one above; entries below a leading entry are zero. Reduced echelon form (RREF): additionally each leading entry is $1$ and is the only nonzero entry in its column.

Theorem 3 (Uniqueness of the reduced echelon form).

Each matrix is row equivalent to exactly one reduced echelon matrix. Echelon form is not unique — RREF is. This distinction is tested directly.

Pivots. A pivot position is the location of a leading 1 in the RREF; its column is a pivot column. Variables in pivot columns are basic; the rest are free. The general solution is written by solving each basic variable in terms of the free ones.

Tested as

Classify matrices as echelon / reduced echelon — 1.2.2
Row reduce and identify pivots — 1.2.3, 1.2.4
General solution from an augmented matrix — 1.2.8, 1.2.9, 1.2.11, 1.2.13, 1.2.14
Consistency from pivot counts — 1.2.35, 1.2.36

Trap

The system is inconsistent exactly when the augmented (last) column is a pivot column. A free variable means infinitely many solutions only if the system is consistent first.

§1.3 Vector Equations

Definition 4 (Linear combination and Span).

Given $\mathbf{v}_1,\dots,\mathbf{v}_p$ and scalars $c_1,\dots,c_p$, the vector $c_1\mathbf{v}_1+\cdots+c_p\mathbf{v}_p$ is a linear combination. $\text{Span}\{\mathbf{v}_1,\dots,\mathbf{v}_p\}$ is the set of all such combinations.

The central translation

These three questions are the same question:

$$\mathbf{b} \in \text{Span}\{\mathbf{a}_1,\dots,\mathbf{a}_n\} \iff x_1\mathbf{a}_1+\cdots+x_n\mathbf{a}_n=\mathbf{b} \text{ is consistent} \iff [\,\mathbf{a}_1\ \cdots\ \mathbf{a}_n \mid \mathbf{b}\,] \text{ is consistent}$$

Nearly every §1.3 problem is an application of this equivalence.

Geometry. $\text{Span}\{\mathbf{v}\}$ is a line through the origin; $\text{Span}\{\mathbf{u},\mathbf{v}\}$ with $\mathbf{u},\mathbf{v}$ not multiples is a plane through the origin. Every span contains $\mathbf{0}$.

Tested as

Vector arithmetic — 1.3.1
Convert between vector equations and systems — 1.3.5, 1.3.6, 1.3.9
Is $\mathbf{b}$ a linear combination? — 1.3.11, 1.3.13, 1.3.14
Solve for a parameter $h$ so $\mathbf{b}$ lies in a plane — 1.3.17
Center of mass / physical application — 1.3.37
True/false on span notation — 1.3.23 – 1.3.33

§1.4 The Matrix Equation $A\mathbf{x}=\mathbf{b}$

Definition 5 (Definition of $A\mathbf{x}$).

For $A$ with columns $\mathbf{a}_1,\dots,\mathbf{a}_n$ and $\mathbf{x}\in\mathbb{R}^n$:

$$A\mathbf{x}=x_1\mathbf{a}_1+x_2\mathbf{a}_2+\cdots+x_n\mathbf{a}_n$$

$A\mathbf{x}$ is defined only when the number of columns of $A$ equals the number of entries of $\mathbf{x}$.

Theorem 6 (Theorem 4 — the four equivalent statements).

For an $m\times n$ matrix $A$, these are all true or all false together:

For each $\mathbf{b}\in\mathbb{R}^m$, $A\mathbf{x}=\mathbf{b}$ has a solution
Each $\mathbf{b}\in\mathbb{R}^m$ is a linear combination of the columns of $A$
The columns of $A$ span $\mathbb{R}^m$
$A$ has a pivot position in every row

Note this is about rows, not columns — a very common slip.

Tested as

Compute $A\mathbf{x}$ two ways — 1.4.1, 1.4.4
Translate system ↔ vector equation ↔ matrix equation — 1.4.6, 1.4.8, 1.4.9, 1.4.10
Does $A\mathbf{x}=\mathbf{b}$ have a solution for every $\mathbf{b}$? — 1.4.15, 1.4.17
Do given vectors span $\mathbb{R}^3$? — 1.4.21, 1.4.22, 1.4.42

Trap

Three vectors in $\mathbb{R}^4$ can never span $\mathbb{R}^4$ — you cannot get a pivot in all four rows from three columns. Spanning needs a pivot in every row; independence needs a pivot in every column.

§1.5 Solution Sets of Linear Systems

Definition 7 (Homogeneous vs. nonhomogeneous).

$A\mathbf{x}=\mathbf{0}$ is homogeneous and is always consistent — $\mathbf{x}=\mathbf{0}$ (the trivial solution) always works. It has a nontrivial solution $\iff$ the system has at least one free variable.

Parametric vector form

Solutions of $A\mathbf{x}=\mathbf{0}$ are written $\mathbf{x}=s\mathbf{u}+t\mathbf{v}$ — a span through the origin. Solutions of a consistent $A\mathbf{x}=\mathbf{b}$ are $\mathbf{x}=\mathbf{p}+\mathbf{v}_h$ where $\mathbf{p}$ is one particular solution and $\mathbf{v}_h$ ranges over solutions of the homogeneous equation.

Geometrically: the solution set of $A\mathbf{x}=\mathbf{b}$ is the solution set of $A\mathbf{x}=\mathbf{0}$ translated by $\mathbf{p}$. It passes through the origin only when $\mathbf{b}=\mathbf{0}$.

Tested as

Nontrivial solutions? — 1.5.1, 1.5.2, 1.5.3
Parametric vector form — 1.5.8, 1.5.9, 1.5.10
Line through $\mathbf{a}$ parallel to $\mathbf{b}$ — 1.5.23
Pivot-count reasoning — 1.5.41, 1.5.42, 1.5.44

§1.7 Linear Independence

Definition 8 (Linear independence).

$\{\mathbf{v}_1,\dots,\mathbf{v}_p\}$ is linearly independent if $x_1\mathbf{v}_1+\cdots+x_p\mathbf{v}_p=\mathbf{0}$ forces $x_1=\cdots=x_p=0$. Equivalently: the columns of $A$ are independent $\iff$ $A\mathbf{x}=\mathbf{0}$ has only the trivial solution $\iff$ $A$ has a pivot in every column.

Independence by inspection

Several problems are answerable with no row reduction at all:

A set containing the zero vector is always dependent
Two vectors are dependent $\iff$ one is a scalar multiple of the other
More vectors than entries ($p>n$ in $\mathbb{R}^n$) forces dependence

These cover 1.7.15 – 1.7.19 outright.

Tested as

Independence of a vector set — 1.7.2, 1.7.4
Independence of matrix columns — 1.7.5 – 1.7.8
By inspection — 1.7.15 – 1.7.19
Pivot counts for independence vs. spanning — 1.7.33, 1.7.34
Build a dependence relation from a stated column relation — 1.7.37, 1.7.38

The distinction to memorize

Spanning ⇒ pivot in every row. Independence ⇒ pivot in every column. For a square matrix these coincide; otherwise they are genuinely different conditions, and the true/false problems exploit exactly that confusion.

Chapter 1 in one page

Question	Condition on $A$
Is $A\mathbf{x}=\mathbf{b}$ consistent?	last column of $[A\mid\mathbf{b}]$ is not a pivot column
Solution unique?	no free variables (pivot in every column)
Do columns span $\mathbb{R}^m$?	pivot in every row
Are columns independent?	pivot in every column
Nontrivial solution to $A\mathbf{x}=\mathbf{0}$?	at least one free variable

Graph

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